= Outgoing shell source estimate for a radial wave
In three dimensions set $w=ru$ for a <radial function> satisfying $\Box u=F$. If the source has <support> in $s\leq r\leq s+1$ and $|F(s,r)|\leq(1+s)^{-2}$, the <Duhamel principle> bounds its contribution to $w(t,t+a)$ by $\frac12\log(1+t)$ for $1/2\leq a\leq1$. Its backward characteristic interval meets the source in a set of length at most one, and $r|F|\leq(1+s)^{-1}$. Division by $r\simeq1+t$ gives $|u|\lesssim\log(2+t)/(1+t)$ for compactly supported initial data.
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